The Schrödinger operator $T = (i\nabla +b)^2+a \cdot \nabla + q$ on
$\mathbb{R}^N$ is considered for $N \ge 2$. Here $a=(a_{j})$ and
$b=(b_{j})$ are real-vector-valued functions on $\mathbb{R}^N$,
while $q$ is a complex-scalar-valued function on $\mathbb{R}^N$.
Over twenty years ago late Professor Kato proved that the minimal
realization $T_{min}$ is essentially quasi-$m$-accretive in
$L^2(\mathbb{R}^N)$ if, among others, $(1+|x|)^{-1}a_j \in
L^4(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$. In this paper it is
shown that under some additional conditions the same conclusion
remains true even if $a_j \in L^4_{loc}(\mathbb{R}^N)$.